Question: Simplify; express your answer in exponential form. Assume $q\neq 0, t\neq 0$. $\dfrac{{(qt^{2})^{2}}}{{(q^{2}t^{-2})^{2}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(qt^{2})^{2} = (q)^{2}(t^{2})^{2}}$ On the left, we have ${q}$ to the exponent ${2}$ . Now ${1 \times 2 = 2}$ , so ${(q)^{2} = q^{2}}$ Apply the ideas above to simplify the equation. $\dfrac{{(qt^{2})^{2}}}{{(q^{2}t^{-2})^{2}}} = \dfrac{{q^{2}t^{4}}}{{q^{4}t^{-4}}}$ Break up the equation by variable and simplify. $\dfrac{{q^{2}t^{4}}}{{q^{4}t^{-4}}} = \dfrac{{q^{2}}}{{q^{4}}} \cdot \dfrac{{t^{4}}}{{t^{-4}}} = q^{{2} - {4}} \cdot t^{{4} - {(-4)}} = q^{-2}t^{8}$